Using SO or SJ cord for permanent wiring of the receptacle in a drawer is not permitted as per Art 400.
It also does not satisify the requirement for a bathroom receptacls as per 210.52, if this is installed in the bathroom.
I disagree, as mentioned in the thread linked to on the first page from about a month ago. I do not see it as a substitute for the fixed wiring of a structure, I don't see it as premises wiring at all. Just my opinion, that and 50 cents will buy you a cup of gas station coffee, if you bring your own cup.
I'm gonna see if I can persuade the folks who own the house I did this in, to take a picture or two of it and the cord. It's been in daily service for probably a year and a half by this point, perhaps if they indulge us we can see if there is any damage to the cord.
I have a sugestion that may help. I have installed a plug in the under-sink area of a vanity for the same purpose as you state. The advantage is the recepticle is in a fixed spot and the hair dryer and curling iron can hang on the vanity door when not used. the cords should be long enough to use them and there will be more space in the drawers for clutter.
I had the same thing in one of my custom homes what i did was i installed a wire mold box behind a flip down drawer like a sponge drawer and they could just pull there hair dryer out. The home owner was happy